Chapter 8: Kinematics of Linear Motion

8.1 Displacement, Velocity and Acceleration as a Function of Time

• Displacement, s, of a particle from a fixed point is its distance from that point in a given direction; it is a vector quantity. (p.252)
• Distance is a scalar quantity: the total path travelled. (p.252)
• s < 0: particle is to the left of O; s = 0: at O; s > 0: to the right of O. (p.253)
• Velocity is the rate of change of displacement with time; it is a vector quantity. (p.254)
• Instantaneous velocity is the gradient of the tangent to the displacement–time graph at that time. (p.255)
• v > 0: particle moves to the right; v = 0: at instantaneous rest; v < 0: moves to the left. (p.255)
• Acceleration is the rate of change of velocity with time; it is a vector quantity. (p.256)
• Instantaneous acceleration is the gradient of the tangent to the velocity–time graph. (p.256)
• A negative acceleration means the particle is decelerating. (p.256)
• a > 0: velocity increasing; a = 0: velocity is maximum or minimum; a < 0: velocity decreasing. (p.257)
• Total distance travelled: sum of the distances found from the number line or displacement–time graph (not the same as the displacement). (p.258)
• Draw a number line showing positions at t = 0, at each time when v = 0, and at the end time. (p.269)

8.2 Differentiation in Kinematics of Linear Motion

• Velocity function: v = ds/dt. (p.260)
• Acceleration function: a = dv/dt = d²s/dt². (p.260)
• Maximum or minimum displacement occurs when ds/dt = v = 0. (p.263)
• A linear displacement–time graph has constant gradient, so the velocity is constant. (p.263)
• A particle moves to the left when v < 0; solve the inequality using the sign of the quadratic. (p.264)
• For a linear velocity–time graph, the acceleration is constant (uniform acceleration). (p.264)
• On the velocity–time graph, a = dv/dt > 0 where v increases; a < 0 (deceleration) where v decreases; a = 0 at maximum velocity. (p.265)
• Zero acceleration does not mean zero velocity. (p.265)

8.3 Integration in Kinematics of Linear Motion

• Velocity from acceleration: v = ∫ a dt. (p.267)
• Displacement from velocity: s = ∫ v dt. (p.269)
• From acceleration, integrate twice: v = ∫ a dt, then s = ∫ v dt. (p.269)
• Use the initial conditions (e.g. t = 0 with a given v or s) to find the constant of integration. (p.267)
• Minimum or maximum velocity occurs when dv/dt = a = 0: d²v/dt² > 0 gives minimum; < 0 gives maximum. (p.268)
• Total distance in the first n seconds is from t = 0 to t = n; distance in the nth second is |s_n − s_n−1|. (p.270)
• Time is a scalar quantity, so its value must be positive. (p.270)

8.4 Applications of Kinematics of Linear Motion

• Using differentiation: v = ds/dt, a = dv/dt. Using integration: v = ∫ a dt, s = ∫ v dt. (p.272)
• Distance travelled can be checked as the definite integral of velocity, ∫₀^t v dt (area under the velocity–time graph). (p.273)